Unit 3: Functions and Patterns - The Monterey Institute for

Transcription

Algebra 1—An Open Course
Professional Development
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Unit 3 – Table of Contents
Unit 3: Functions and Patterns
Video Overview
Learning Objectives
3.2
Media Run Times
3.3
Instructor Notes
• The Mathematics of Functions and Patterns
• Teaching Tips: Conceptual Challenges and Approaches
3.4
Instructor Overview
• Tutor Simulation: Snowboarding
3.7
Instructor Overview
• Puzzle: What Comes Next?
3.8
Instructor Overview
• Project: Design a Rollercoaster
3.9
Glossary
3.13
Common Core Standards
3.15
Some rights reserved.
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Monterey Institute for Technology and Education 2011 V1.1
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Unit 3 – Learning Objectives
Objectives
Lesson 1: Working with Patterns
Topic 1: Inductive Patterns
Learning Objectives
• Recognize patterns in sequences of numbers and geometric/real world
objects.
• Apply inductive reasoning to predict the value of unknown terms in a
sequence.
Topic 2: Representing Patterns
Learning Objectives
• Use tables and graphs to identify and represent patterns.
Lesson 2: Graphing Functions and Relations
Topic 1: Representing Functions and Relations
Learning Objectives
• Define, compare, and recognize relations and functions.
• Represent relations and functions with graphs, tables, and sets of ordered
pairs.
Topic 2: Domain and Range
Learning Objectives
• Define domain and range.
• Identify the domain and range for relations described with words, symbols,
tables, sets of ordered pairs, and graphs.
Topic 3: Proportional Functions
Learning Objectives
• Define proportional function.
• Explain the parts of the standard proportional function equation.
• Recognize and describe the characteristics of a proportional function graph.
Topic 4: Linear Functions
Learning Objectives
• Define linear functions and describe their characteristics.
• Compare and contrast proportional and non-proportional linear functions.
• Explain the components of the linear function equation.
Topic 5: Non-linear Functions
Learning Objectives
• Define nonlinear function.
• Define inverse functions and recognize them in equation, table, and graph
form.
• Define quadratic functions and recognize them in equation, table, and
graph form.
• Define exponential functions and recognize them in equation, table, and
graph form.
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Unit 3 - Media Run Times
Unit 3
Lesson 1
Topic 1, Presentation – 4.1 minutes
Topic 1, Worked Example 1 – 2.5 minutes
Lesson 2
Topic 5, Worked Example 1 – 4 minutes
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Unit 3 – Instructor Notes
Instructor Notes
The Mathematics of Functions and Patterns
Unit 3 introduces the idea that the world is full of patterns, patterns that can be found and
described mathematically. Students will learn to recognize patterns within and between
sequences—in groups of objects, in series of numbers, and in tables and graphs of inputs and
outputs. By the time they complete the unit, students will know the definitions and
characteristics of proportional, linear, non-linear, inverse, quadratic, and exponential functions.
This unit contains what may appear to be an overwhelming list of objectives:
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Recognize patterns in sequences of numbers and geometrical/real world objects.
Apply inductive reasoning to predict the value of unknown terms in a sequence.
Use tables and graphs to identify and represent patterns.
Define, compare, and recognize relations and functions.
Represent relations and functions with graphs, tables, and sets of ordered pairs.
Define domain and range.
Identify the domain and range for relations described with words, symbols, tables, sets
of ordered pairs, and graphs.
Define proportional function.
Explain the parts of the standard proportional function equation.
Recognize and describe the characteristics of a proportional function graph.
Define linear functions and describe their characteristics.
Compare and contrast proportional and non-proportional linear functions.
Explain the components of the linear function equation.
Define nonlinear function.
Define inverse functions and recognize them in equation, table, and graph form.
Define quadratic functions and recognize them in equation, table, and graph form.
Define exponential functions and recognize them in equation, table, and graph form.
But most of these concepts are only introduced and not explored in depth. For example, the last
objective, “define exponential functions and recognize them in equation, table, and graph form”
requires that students recognize the general form of an exponential function, but not that they
solve one for unknowns. The applications and complexities of functions will generally be left for
Algebra 2 and other courses.
Teaching Tips: Conceptual Challenges and Approaches
Students will need to make and use multiple representations of functions and relations,
including words, tables, graphs, pictures, and symbolic algebra. The ability to translate
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comfortably between these media and understand how to extract information from each context
underlies much of a student’s potential success in Algebra 1, and beyond.
Allowing students time to generate their own different representations of functions is a critical
need. For example, leading students though examples of the typical equation ! table ! graph
relationship is important initially. Once students understand the concept, it is then critical that
they have time to develop their own methods for pulling information out of data. This is probably
best done in a small group setting, with the teacher facilitating as students explore proportional
relationships or patterns.
Example
The example used within the video presentation of Lesson 2 Topic 1 illustrates the kind of
scenario that works well for this type of semi-independent work. Students could begin with the
situation from “Just Bicycles”:
Have them create tables, graphs, verbal descriptions, and an equation to describe the
relationship between the number of bicycles in the store and the number of wheels.
Then students could consider the case of "Multi-Cycles":
Once again, ask them to develop tables, graphs, verbal descriptions, and an equation (or
equations) to describe the relationship between the number of cycles in the store and the
number of wheels. When they realize they can't write a simple equation for this relationship, it
sets up a discussion of the differences between functions and relations. Many other key topics
for the unit, such as domain and range, are also easily developed out of this type of scenario.
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Hands-on Opportunities
The text of this unit includes 3 manipulatives that lets students to tinker with the graphs of nonlinear functions. These are:
• Quadratic Functions (Lesson 2 Topic 5)
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In this manipulative, the values of a, b, and c in the standard quadratic formula y = ax2 +
bx + c can all be altered. Students will see how the shape and orientation of a parabola
is affected by changing these variables.
Exponential Functions (Lesson 2 Topic 5)
In this manipulative, the coefficients a and b in the exponential function equation y = abx
can be changed with slider bars. Students will notice the large effect of small changes on
the shape of the graph.
As students create tables and graphs and work with the manipulatives, teachers should ask
questions to nudge them into understanding and exploring the meaning of inputs and outputs,
relations and functions. Students who spend time discussing these ideas are more likely to
understand what they really mean.
Teaching Tips: Algorithmic Challenges and Approaches
This unit uses relatively few algorithms, and most of those are very simple, so few students will
have trouble with calculations or procedures. However, the ability to accurately draw and
interpret graphs is very important. There are aspects of working with graphs that deserve
particular focus:
• Students must use rulers to draw graphs and understand that marks on the axes need to
be evenly spaced (This can be aided by the use of graph paper in the early stages).
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Appropriate graphs (and/or tables of data) from newspapers or from Internet news
articles can be a powerful way of helping students see the connection between the
classroom and the world. These examples can also be used to emphasize how reading
data incorrectly from a graph might have significant consequences.
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Real-world scenarios can be used to further develop the ideas of function vs. relation,
range, domain, and ordered pairs. For example, students could compile the set of
people in the class and their heights.
Summary
This unit begins the process of helping students visualize and think about numbers, shapes and
situations in terms of mathematical patterns. It also introduces the idea of a function, which will
be developed more fully in Unit 4: Analyze and Graph Linear Equations, Functions and
Relations and Unit 10: Quadratic Functions.
With lots of hands-on practice creating graphs and charts of inputs and outputs, it is possible to
present what at first seems like a daunting list of objectives as a coherent set of interlinked
ideas.
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Unit 3 – Tutor Simulation
Instructor Overview
Tutor Simulation: Snowboarding
Purpose
This simulation allows students to demonstrate their understanding of functions and relations.
Students will be asked to apply what they have learned to solve a real-world problem involving:
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Patterns
Functions
Relations
Domain and range
Problem
Students are given the following problem:
We're sending you on a simulated trip to the mountains to go snowboarding. When you get
there, you'll notice that the different runs down the mountain--and the lift to the top--all look
familiar. Your challenge will be to analyze the different paths up and down the mountain and
determine if they are functions and/or relations, linear or non-linear, and
proportional or non-proportional.
Next, you'll take a closer look at two of the paths, and use tables and sets of ordered pairs to
determine the height to distance rate of change of these paths. Finally you'll
develop equations that describe the runs.
Recommendations
Tutor simulations are designed to give students a chance to assess their understanding of unit
material in a personal, risk-free situation. Before directing students to the simulation,
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make sure they have completed all other unit material.
explain the mechanics of tutor simulations
o Students will be given a problem and then guided through its solution by a video
tutor;
o After each answer is chosen, students should wait for tutor feedback before
continuing;
o After the simulation is completed, students will be given an assessment of their
efforts. If areas of concern are found, the students should review unit materials or
seek help from their instructor.
emphasize that this is an exploration, not an exam.
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Unit 3 – Puzzle
Instructor Overview
Puzzle: What Comes Next?
Objective
What Comes Next? is a puzzle that presents a pattern of colored shapes and invites the learner
to complete the sequence. Players must recognize the pattern in a series of colored geometric
shapes, and apply inductive reasoning to predict the next terms in the sequence.
Figure 1. What Comes Next? asks players to first determine and then continue the pattern created by a
sequence of colored shapes.
Description
In each level, players begin with a sequence of colored shapes, and must pick from sets of
more shapes to continue the pattern. Learners are rewarded for completing the sequences
correctly, and the faster they do it the more lavish the praise. There are three levels of
difficulty—the easy level has a 3 shape pattern, the medium level starts with a 5 piece pattern,
and the difficult level has 7 shapes in the initial sequence.
There are ten set sequences at each level, so the game play is not endless. However, the more
difficult the patterns the less likely the inferences necessary to complete the sequence will be
made from memory than afresh. What Comes Next? is suitable for solo play, for two or more
players, or even in a classroom setting where learners could call out the pairs by color or shape
or both.
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Unit 3 – Project
Instructor Overview
Project: Design a Rollercoaster
Student Instructions
Introduction
Almost everyone has ridden or at least seen a roller coaster in action. Did you know that there is
a connection between roller coasters and the algebra you have been studying in this unit?
Engineers rely on their knowledge of algebra to design roller coasters to make them both fun
and scary at the same time. Use these websites to help you study the algebra involved in roller
coaster designs.
http://www.funderstanding.com/coaster
http://mathdl.maa.org/images/upload_library/4/vol5/coaster/coasterapplet.htm
http://nlvm.usu.edu/en/nav/frames_asid_331_g_3_t_2.html?from=category_g_3_t_2.html
Task
For this project you or your team will use your knowledge of algebra functions and patterns to
determine how engineers use their knowledge of math to design roller coasters. You will draw a
picture of roller coaster and plot ordered pairs along the path a coaster car travels on the
coaster’s rails. You will also compare these ordered points by constructing a functions table and
a graph to plot the ordered points. You can use the websites to help explore the design of roller
coasters, identifying ordered points on a coaster, and review how to plot ordered points on a
graph.
Instructions
Draw the side view of a roller coaster of your choice or one of your own design on a piece of
poster board. The roller coaster should have at least 2 hills and 1 loop. Layout both the “X” and
“Y” axis, labeling each axis with appropriate intervals for selecting ordered pairs. Select ordered
pairs, one pair at interval points of your choice on the drawing along the track of your roller
coaster. Record these ordered points in a function table and then create a graph of these
values.
Problem 1:
Write a 1 – 2 page paper that compares and contrasts how your roller coaster design relates to
proportional function. Include a graph and table to support your discussion.
Problem 2:
Write a 1 – 2 page paper that compares and contrasts how your roller coaster design relates to
linear or non-linear function. Include a graph and table to support your discussion.
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Problem 3:
Create an interactive poster using Glogster to explain why or why not your roller coaster has a
relationship between its design and functions in algebra.
Collaboration
Discuss with other students or team members the relationship between your roller coaster and
the design of other roller coasters. Use pictures of roller coasters at Hershey Park or Six Flags
Great America for comparison at the following websites:
http://www.coastergallery.com/2000T/hp.html
http://www.coastergallery.com/1999T/SFGA.html
Conclusions
Share the findings of your project with others using any of the following methods:
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Present the answers to the problems and interactive poster to the other students in your
class.
Present the answers to the problems and interactive poster to the other students in your
class using a multimedia presentation.
Post the answers on the class Wiki for sharing with parents and fellow students. Request
feedback and comments from those view your work.
Instructor Notes
Assignment Procedures
Problem 1
This task focuses on use of sets of ordered pairs (taken from the roller coaster design) to
recognize a proportional function. Instruct students that their description should include a
discussion of whether their roller coaster is proportional or non-proportional, along with
appropriate formula(s) to support their findings.
Problem 2
This question asks students to recognize and compare the linear and non-linear portions of their
roller coaster. Instruct them to explain their reasoning, along with the appropriate formula(s) to
support their findings.
Problem 3
This question requires students to make an interactive poster of their findings in earlier
problems. Look up Glogster (http://edu.glogster.com) as a class and make sure everyone
understands how to find and use the site.
Suggest that students incorporate video clips to illustrate their project. Videos of roller coasters
are available on YouTube, or students can create one at http://www.animoto.com/education.
Recommendations:
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have students work in teams to encourage brainstorming and cooperative learning.
assign a specific timeline for completion of the project that includes milestone dates.
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provide students feedback as they complete each milestone.
ensure that each member of student groups has a specific job.
Technology Integration
This project provides abundant opportunities for technology integration, and gives students the
chance to research and collaborate using online technology. The following are examples of free
internet resources that can be used to support this project:
http://www.moodle.org
An Open Source Course Management System (CMS), also known as a Learning Management
System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular
among educators around the world as a tool for creating online dynamic websites for their
students.
http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview Lets
you create a secure online Wiki workspace in about 60 seconds. Encourage classroom
participation with interactive Wiki pages that students can view and edit from any computer.
Share class resources and completed student work with parents.
http://www.docs.google.com
Allows students to collaborate in real-time from any computer. Google Docs provides free
access and storage for word processing, spreadsheets, presentations, and surveys. This is
ideal for group projects.
http://why.openoffice.org/
The leading open-source office software suite for word processing, spreadsheets,
presentations, graphics, databases and more. It can read and write files from other common
office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded
and used completely free of charge for any purpose.
Rubric
Score
4
Content
Your project includes a complete
understanding of the algebra functions
required to solve the problems.
Your written reports use appropriate
algebra functions to answer problems.
Presentation
Your presentation is detailed and
clear. Explanation includes the
important algebra functions required
in the project in logical sequence
that is easy to understand.
Your project uses appropriate graphs and
tables to support written reports.
Your interactive poster includes all the
appropriate pictures, video(s), graphs, and
tables.
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Your project includes a good understanding
of the algebra functions required to solve
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Your presentation is clear.
Explanation includes the important
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the problems.
Your written reports use appropriate
algebra functions to answer problems.
algebra functions required in the
project in logical sequence that is
easy to understand.
Your project uses appropriate graphs and
tables to support written reports.
Your poster includes all the appropriate
pictures, video(s), graphs, and tables.
However, it is not interactive.
2
Your project includes a good understanding
of the algebra functions required to solve
the problems.
Your written reports do not include all the
appropriate algebra functions to answer
problems.
Your presentation is a little difficult
to understand. However, it does
include most of the important
algebra functions required in the
project.
Your project’s graphs and tables to support
written reports.
Your poster pictures, video(s), graphs, and
tables do not support the algebra ideas of
this unit. However, it is interactive.
1
Your project includes does not show you
understand the algebra functions required
to solve the problems.
Your written reports does is indicate you
understand all the appropriate algebra
functions to answer problems.
Your project’s written reports do not include
graphs and tables.
You did not complete the interactive poster.
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Your presentation is difficult to
understand and does not follow a
logical thought process. Several
important algebra functions required
in the project are missing.
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Unit 3: Algebra - Functions and Patterns
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Unit 3 – Glossary
Glossary
constant of
proportionality
the constant in a proportional function equation; it describes the ratio or
proportional relationship of the independent and dependent variables—also
called the constant of variation or the rate of change
constant of variation the constant in a proportional function equation; it describes the ratio or
called the rate of change or the constant of proportionality
continuous pattern
a pattern made of uninterrupted or connected values or objects
dependent value
a value or variable that depends upon the independent value
dependent variable
a value or variable that depends upon the independent value
discrete pattern
a pattern made of separate and distinct values or objects
discrete values
values that change in increments (not continuously)
domain
the set of all possible inputs of a function which allow the function to work
exponential function
a nonlinear function in which the independent value is an exponent in the
function, as in y = abx
function
a kind of relation in which one variable uniquely determines the value of another
variable
independent value
a value or variable that changes or can be manipulated by circumstances
independent variable a value or variable that changes or can be manipulated by circumstances
inductive reasoning
a form of logical thinking that makes general conclusions based on specific
situations, inductive reasoning takes the path of observation to generalization to
conjecture
input
the independent variable of a function—input determines output
inverse function
a nonlinear function in which the reciprocal of the independent variable times a
constant equals the dependent variable, as in
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linear function
a function with a constant rate of change and a straight line graph
mathematical
sequence
an ordered list of numbers or objects
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range
the set of all possible outputs of a function
rate of change
the constant in a proportional function equation; it describes the ratio or
called the constant of variation or the constant of proportionality
relation
the relationship between variables that change together
term
a value in a sequence--the first value in a sequence is the 1st term, the second
value is the 2nd term, and so on; a term is also any of the monomials that make
up a polynomial
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Unit 3 – Common Core
NROC Algebra 1--An Open Course
Unit 3
Mapped to Common Core State Standards, Mathematics
Algebra 1 | Functions and Patterns | Working with Patterns | Inductive Patterns
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.MP.
Mathematical Practices
CATEGORY / CLUSTER
7.MP.2.
Reason abstractly and quantitatively.
CATEGORY / CLUSTER
7.MP.7.
Look for and make use of structure.
CATEGORY / CLUSTER
7.MP.8.
Look for and express regularity in repeated reasoning.
STRAND / DOMAIN
CC.8.MP.
CATEGORY / CLUSTER
8.MP.2.
CATEGORY / CLUSTER
8.MP.7.
CATEGORY / CLUSTER
8.MP.8.
STRAND / DOMAIN
CC.MP.
CATEGORY / CLUSTER
MP-2.
CATEGORY / CLUSTER
MP-7.
CATEGORY / CLUSTER
MP-8.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
Interpreting Functions
Grade: 9-12 - Adopted 2010
STANDARD
EXPECTATION
Understand the concept of a function and use function notation.
F-IF.3.
Recognize that sequences are functions, sometimes defined recursively, whose
domain is a subset of the integers. For example, the Fibonacci sequence is
defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n greater than or
equal to 1.
Algebra 1 | Functions and Patterns | Working with Patterns | Representing Patterns
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.3.
Recognize that sequences are functions, sometimes defined recursively, whose
domain is a subset of the integers. For example, the Fibonacci sequence is
defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n greater than or
equal to 1.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-BF.
Building Functions
STANDARD
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Build a function that models a relationship between two quantities.
EXPECTATION
F-BF.1.
Write a function that describes a relationship between two quantities.
GRADE EXPECTATION
F-BF.1.a.
Determine an explicit expression, a recursive process, or steps for calculation
from a context.
STRAND / DOMAIN
CC.F.
Functions
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CATEGORY / CLUSTER
F-BF.
STANDARD
EXPECTATION
Building Functions
F-BF.2.
Write arithmetic and geometric sequences both recursively and with an explicit
formula, use them to model situations, and translate between the two forms.
Algebra 1 | Functions and Patterns | Graphing Functions and Relations | Representing Functions and Relations
STRAND / DOMAIN
CC.7.MP.
CATEGORY / CLUSTER
7.MP.4.
Model with mathematics.
STRAND / DOMAIN
CC.8.MP.
CATEGORY / CLUSTER
8.MP.4.
STRAND / DOMAIN
CC.8.EE.
Expressions and Equations
CATEGORY / CLUSTER
Understand the connections between proportional relationships, lines, and linear
equations.
STANDARD
8.EE.5.
Graph proportional relationships, interpreting the unit rate as the slope of the
graph. Compare two different proportional relationships represented in different
ways. For example, compare a distance-time graph to a distance-time equation
to determine which of two moving objects has greater speed.
STRAND / DOMAIN
CC.8.F.
Functions
CATEGORY / CLUSTER
Define, evaluate, and compare functions.
STANDARD
8.F.1.
Understand that a function is a rule that assigns to each input exactly one
output. The graph of a function is the set of ordered pairs consisting of an input
and the corresponding output.
STRAND / DOMAIN
CC.MP.
CATEGORY / CLUSTER
MP-4.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.1.
Understand that a function from one set (called the domain) to another set
(called the range) assigns to each element of the domain exactly one element of
the range. If f is a function and x is an element of its domain, then f(x) denotes
the output of f corresponding to the input x. The graph of f is the graph of the
equation y = f(x).
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
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Analyze functions using different representations.
EXPECTATION
F-IF.9.
Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For
example, given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
STRAND / DOMAIN
CC.M.
Modeling
CATEGORY / CLUSTER
M-2.
Formulating a model by creating and selecting geometric, graphical, tabular,
algebraic, or statistical representations that describe relationships between the
variables
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Algebra 1 | Functions and Patterns | Graphing Functions and Relations | Domain and Range
STRAND / DOMAIN
CC.7.MP.
CATEGORY / CLUSTER
7.MP.4.
STRAND / DOMAIN
CC.8.MP.
CATEGORY / CLUSTER
8.MP.4.
STRAND / DOMAIN
CC.8.EE.
CATEGORY / CLUSTER
equations.
STANDARD
8.EE.5.
STRAND / DOMAIN
CC.MP.
CATEGORY / CLUSTER
MP-4.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
Interpret functions that arise in applications in terms of the context.
EXPECTATION
F-IF.5.
Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the
number of person-hours it takes to assemble n engines in a factory, then the
positive integers would be an appropriate domain for the function.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.9.
STRAND / DOMAIN
CC.M.
Modeling
CATEGORY / CLUSTER
M-2.
Formulating a model by creating and selecting geometric, graphical, tabular,
algebraic, or statistical representations that describe relationships between the
variables
Algebra 1 | Functions and Patterns | Graphing Functions and Relations | Proportional Functions
STRAND / DOMAIN
CC.7.RP.
CATEGORY / CLUSTER
Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and
mathematical problems.
STANDARD
7.RP.2.
Recognize and represent proportional relationships between quantities.
EXPECTATION
7.RP.2.a.
Decide whether two quantities are in a proportional relationship, e.g., by testing
for equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin.
EXPECTATION
7.RP.2.c.
Represent proportional relationships by equations. For example, if total cost t is
proportional to the number n of items purchased at a constant price p, the
relationship between the total cost and the number of items can be expressed as
t = pn.
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STRAND / DOMAIN
CC.8.EE.
CATEGORY / CLUSTER
equations.
STANDARD
8.EE.5.
STRAND / DOMAIN
CC.8.F.
Functions
CATEGORY / CLUSTER
Use functions to model relationships between quantities.
STANDARD
8.F.4.
Construct a function to model a linear relationship between two quantities.
Determine the rate of change and initial value of the function from a description
of a relationship or from two (x, y) values, including reading these from a table
or from a graph. Interpret the rate of change and initial value of a linear
function in terms of the situation it models, and in terms of its graph or a table
of values.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-CED.
Creating Equations
STANDARD
Create equations that describe numbers or relationships.
EXPECTATION
A-CED.1.
Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
EXPECTATION
A-CED.2.
Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
EXPECTATION
A-CED.3.
Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling context. For example, represent inequalities describing nutritional and
cost constraints on combinations of different foods.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-REI.
Reasoning with Equations and Inequalities
STANDARD
Understand solving equations as a process of reasoning and explain the
reasoning.
EXPECTATION
A-REI.1.
Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the
original equation has a solution. Construct a viable argument to justify a solution
method.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-REI.
STANDARD
Represent and solve equations and inequalities graphically.
EXPECTATION
A-REI.10.
Understand that the graph of an equation in two variables is the set of all its
solutions plotted in the coordinate plane, often forming a curve (which could be
a line).
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
!
STANDARD
EXPECTATION
!
F-IF.4.
For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs
showing key features given a verbal description of the relationship. Key features
include: intercepts; intervals where the function is increasing, decreasing,
positive, or negative; relative maximums and minimums; symmetries; end
behavior; and periodicity.
"#$*!
!
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.7.
Graph functions expressed symbolically and show key features of the graph, by
hand in simple cases and using technology for more complicated cases.
GRADE EXPECTATION
F-IF.7.a.
Graph linear and quadratic functions and show intercepts, maxima, and minima.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.9.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-BF.
Building Functions
STANDARD
EXPECTATION
F-BF.1.
Write a function that describes a relationship between two quantities.
GRADE EXPECTATION
F-BF.1.a.
Determine an explicit expression, a recursive process, or steps for calculation
from a context.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-BF.
Building Functions
STANDARD
Build new functions from existing functions.
EXPECTATION
F-BF.4.
Find inverse functions.
GRADE EXPECTATION
F-BF.4.a.
Solve an equation of the form f(x) = c for a simple function f that has an inverse
and write an expression for the inverse. For example, f(x) =2 x^3 for x > 0 or f(x)
= (x+1)/(x-1) for x not equal to 1.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-LE.
Linear and Exponential Models
STANDARD
Construct and compare linear and exponential models and solve problems.
EXPECTATION
F-LE.1.
Distinguish between situations that can be modeled with linear functions and
with exponential functions.
GRADE EXPECTATION
F-LE.1.a.
Prove that linear functions grow by equal differences over equal intervals, and
that exponential functions grow by equal factors over equal intervals.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-LE.
STANDARD
EXPECTATION
F-LE.2.
Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output
pairs (include reading these from a table).
Algebra 1 | Functions and Patterns | Graphing Functions and Relations | Linear Functions
STRAND / DOMAIN
CC.7.RP.
CATEGORY / CLUSTER
STANDARD
!
Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and
mathematical problems.
7.RP.2.
Recognize and represent proportional relationships between quantities.
"#$+!
!
EXPECTATION
7.RP.2.a.
Decide whether two quantities are in a proportional relationship, e.g., by testing
for equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin.
STRAND / DOMAIN
CC.8.EE.
CATEGORY / CLUSTER
equations.
STANDARD
8.EE.5.
STRAND / DOMAIN
CC.8.EE.
CATEGORY / CLUSTER
Analyze and solve linear equations and pairs of simultaneous linear equations.
STANDARD
8.EE.8.
Analyze and solve pairs of simultaneous linear equations.
EXPECTATION
8.EE.8.a.
Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of
intersection satisfy both equations simultaneously.
EXPECTATION
8.EE.8.b.
Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by inspection.
For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot
simultaneously be 5 and 6.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-CED.
Creating Equations
STANDARD
EXPECTATION
A-CED.2.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-REI.
STANDARD
Solve systems of equations.
EXPECTATION
A-REI.6.
Solve systems of linear equations exactly and approximately (e.g., with graphs),
focusing on pairs of linear equations in two variables.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-REI.
!
STANDARD
A-REI.10.
EXPECTATION
A-REI.11.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
!
EXPECTATION
a line).
Explain why the x-coordinates of the points where the graphs of the equations y
= f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the
solutions approximately, e.g., using technology to graph the functions, make
tables of values, or find successive approximations. Include cases where f(x)
and/or g(x) are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions.
"#%,!
!
EXPECTATION
F-IF.4.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.7.
GRADE EXPECTATION
F-IF.7.a.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-LE.
STANDARD
EXPECTATION
F-LE.1.
GRADE EXPECTATION
F-LE.1.a.
Algebra 1 | Functions and Patterns | Graphing Functions and Relations | Non-linear Functions
STRAND / DOMAIN
CC.8.F.
CATEGORY / CLUSTER
Functions
Define, evaluate, and compare functions.
STANDARD
8.F.2.
example, given a linear function represented by a table of values and a linear
function represented by an algebraic expression, determine which function has
the greater rate of change.
STANDARD
8.F.3.
Interpret the equation y = mx + b as defining a linear function, whose graph is a
straight line; give examples of functions that are not linear. For example, the
function A = s^2 giving the area of a square as a function of its side length is not
linear because its graph contains the points (1,1), (2,4) and (3,9), which are not
on a straight line.
STRAND / DOMAIN
CC.8.F.
Functions
CATEGORY / CLUSTER
Use functions to model relationships between quantities.
STANDARD
8.F.5.
Describe qualitatively the functional relationship between two quantities by
analyzing a graph (e.g., where the function is increasing or decreasing, linear or
nonlinear). Sketch a graph that exhibits the qualitative features of a function
that has been described verbally.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-SSE.
Seeing Structure in Expressions
STANDARD
!
Write expressions in equivalent forms to solve problems.
EXPECTATION
A-SSE.3.
Choose and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression.
GRADE EXPECTATION
A-SSE.3.c.
Use the properties of exponents to transform expressions for exponential
functions. For example the expression 1.15^t can be rewritten as
(1.15^(1/12))^12t approximately equals 1.012^12t to reveal the approximate
equivalent monthly interest rate if the annual rate is 15%.
STRAND / DOMAIN
CC.A.
Algebra
"#%$!
!
CATEGORY / CLUSTER
A-CED.
STANDARD
EXPECTATION
A-CED.2.
STRAND / DOMAIN
CC.A.
Algebra
CATEGORY / CLUSTER
A-REI.
STANDARD
EXPECTATION
A-REI.10.
STRAND / DOMAIN
CC.F.
a line).
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.4.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.7.
GRADE EXPECTATION
F-IF.7.a.
GRADE EXPECTATION
F-IF.7.e.
Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and amplitude.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-IF.
STANDARD
EXPECTATION
F-IF.8.
Write a function defined by an expression in different but equivalent forms to
reveal and explain different properties of the function.
GRADE EXPECTATION
F-IF.8.b.
Use the properties of exponents to interpret expressions for exponential
functions. For example, identify percent rate of change in functions such as y =
(1.02)^t, y = (0.97)^t, y = (1.01)^12t, y = (1.2)^t/10, and classify them as
representing exponential growth or decay.
STRAND / DOMAIN
CC.F.
Functions
CATEGORY / CLUSTER
F-LE.
STANDARD
!
Creating Equations
EXPECTATION
F-LE.1.
GRADE EXPECTATION
F-LE.1.a.
GRADE EXPECTATION
F-LE.1.b.
Recognize situations in which one quantity changes at a constant rate per unit
interval relative to another.
GRADE EXPECTATION
F-LE.1.c.
Recognize situations in which a quantity grows or decays by a constant percent
rate per unit interval relative to another.
STRAND / DOMAIN
CC.F.
Functions
"#%%!
!
CATEGORY / CLUSTER
F-LE.
STANDARD
EXPECTATION
F-LE.3.
Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more
generally) as a polynomial function.
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"#%"!

Unit 3: Functions and Patterns - The Monterey Institute for

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